where \(E^{<->}\)  and \(E^{<+>}\)  respectively indicate the set of the edges with negative and with positive weights. The approximation scheme in [1]} reduces Min Cut to Correlation Clustering in order to obtain a logarithmic approximation factor for Correlation Clustering. It also develops a reduction from Correlation Clustering to Min Cut to conclude the equivalence of these two cost functions.
Here, we elaborate that these two cost functions are identical and represent the same objective (given the shifted pairwise similarities) in a direct and straightforward way without using the more complicated reduction argument. In addition, [1]} assumes that the number of clusters is hidden in the cost function (as defined in Eq. REF ). However, we study the equivalence for any arbitrary number of clusters \(K\) . As shown in [3]}, [4]}, optimizing Correlation Clustering without a constraint on the number of clusters can lead to overfitting and unrobust solutions, whereas fixing the number of clusters may avoid these issues. Therefore, we consider the setting where the number of clusters \(K\)  is explicitly specified in the cost function and the user has the possibility to fix it in advance. Finally, the reduction-based argument in [1]} yields the equivalence of the optimal solutions between Min Cut and Correlation Clustering and the respective approximation and hardness results. We, in addition, conclude the equivalence of any local optimal solution for the two cost functions, which is important when using local search algorithms to optimize the cost functions.
